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A213742 Triangle of numbers C^(3)(n,k) of combinations with repetitions from n different elements over k for each of them not more than three appearances allowed. 7
1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 10, 20, 31, 1, 5, 15, 35, 65, 101, 1, 6, 21, 56, 120, 216, 336, 1, 7, 28, 84, 203, 413, 728, 1128, 1, 8, 36, 120, 322, 728, 1428, 2472, 3823, 1, 9, 45, 165, 486, 1206, 2598, 4950, 8451, 13051, 1, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The left side of triangle consists of 1's, while the right side is formed by A005725. Further, T(n,0)=1, T(n,1)=n, T(n,2)=A000217(n) for n>1, T(n,3)=A000292(n) for n>=3, T(n,4)=A005718(n) for n>=2, T(n,5)=A005719(n) for n>=5, T(n,6)=A005720(n) for n>=6, T(n,7)=A001919(n) for n>=7, T(n,8)=A064055(n) for n>=5.

LINKS

Peter J. C. Moses, Rows n = 0..50 of triangle, flattened

FORMULA

C^(3)(n,k)=sum{r=0,...,floor(k/4)}(-1)^r*C(n,r)*C(n-4*r+k-1, n-1)

EXAMPLE

Triangle begins

n/k.|..0.....1.....2.....3.....4.....5.....6.....7

==================================================

.0..|..1

.1..|..1.....1

.2..|..1.....2.....3

.3..|..1.....3.....6....10

.4..|..1.....4....10....20....31

.5..|..1.....5....15....35....65....101

.6..|..1.....6....21....56...120....216...336

.7..|..1.....7....28....84...203....413...728....1128

MATHEMATICA

Flatten[Table[Sum[(-1)^r Binomial[n, r] Binomial[n-# r+k-1, n-1], {r, 0, Floor[k/#]}], {n, 0, 15}, {k, 0, n}]/.{0}->{1}]&[4] (* Peter J. C. Moses, Apr 16 2013 *)

CROSSREFS

Cf. A007318, A005725, A111808, A000217, A000292, A005718, A005719, A005720, A001919, A064055.

Sequence in context: A111808 A247046 A081422 * A213743 A213744 A213745

Adjacent sequences:  A213739 A213740 A213741 * A213743 A213744 A213745

KEYWORD

nonn,tabl

AUTHOR

Vladimir Shevelev and Peter J. C. Moses, Jun 19 2012

STATUS

approved

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Last modified November 28 07:52 EST 2021. Contains 349401 sequences. (Running on oeis4.)